How to maximize any of the views in quad view?
The quad view exists so that you can see all the views at once.

Instead of thinking of it like maximizing one of the quad view windows, think of it like aligning the user controlled view to an axis. 


Numpad 1 snaps user controlled view to front view
Numpad 3 snaps user controlled view to side view
Numpad 7 snaps user controlled view to top view


Holding &#9096; Ctrl will snap to the opposite direction of the axis, so &#9096; CtrlNumpad 7 snaps to bottom view.

Numpad 5 toggles perspective/orthographic view.

Also see the wiki for more information on navigating the 3D view.